Exploring implications of Trace (Inversion) formula and Artin algebras in extremal combinatorics

Abstract

This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian K-algebras (mainly through the Trace Formula), where K is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite \(\mathbb {Q}\)-algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null t-designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341–345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21–23, 1983) of the Sauer–Shelah–Perles Lemma, we produce a new one based only in duality in the \(\mathbb {Q}\)-algebra \(\mathbb {Q}[V_n]\) of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set \([n]:=\{1,2,\ldots , n\}\). All results are equally true if we replace \(\mathbb {Q}[V_n]\) by \(K[V_n]\), where K is any perfect field of characteristics \(\not =2\). The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework.